Delayed Optimal Control of Stochastic LQ Problem

A discrete-time stochastic LQ problem with multiplicative noises and state transmission delay is studied in this paper, which does not require any definiteness constraint on the cost weighting matrices. From some abstract representations of the system and cost functional, the solvability of this LQ problem is characterized by some conditions with operator form. Based on these, necessary and sufficient conditions are derived for the case with a fixed time-state initial pair and the general case with all the time-state initial pairs. For both cases, a set of coupled discrete-time Riccati-like equations can be derived to characterize the existence and the form of the delayed optimal control. In particular, for the general case with all the initial pairs, the existence of the delayed optimal control is equivalent to the solvability of the Riccati-like equations with some algebraic constraints, and both of them are also equivalent to the solvability of a set of coupled linear matrix equality-inequalities. Note that both the constrained Riccati-like equations and the linear matrix equality-inequalities are introduced for the first time in the literature for the proposed LQ problem. Furthermore, the convexity and the {uniform convexity} of the cost functional are fully characterized via certain properties of the solution of the Riccati-like equations

Stabilization Control for ItO Stochastic System with Indefinite State and Control Weight Costs

In standard linear quadratic (LQ) control, the first step in investigating infinite-horizon optimal control is to derive the stabilization condition with the optimal LQ controller. This paper focuses on the stabilization of an Ito stochastic system with indefinite control and state weighting matrices in the cost functional. A generalized algebraic Riccati equation (GARE) is obtained via the convergence of the generalized differential Riccati equation (GDRE) in the finite-horizon case. More importantly, the necessary and sufficient stabilization conditions for indefinite stochastic control are obtained. One of the key techniques is that the solution of the GARE is decomposed into a positive semi-definite matrix that satisfies the singular algebraic Riccati equation (SARE) and a constant matrix that is an element of the set satisfying certain linear matrix inequality conditions. Using the equivalence between the GARE and SARE, we reduce the stabilization of the general indefinite case to that of the definite case, in which the stabilization is studied using a Lyapunov functional defined by the optimal cost functional subject to the SARE.Comment: 8 pages, 3 figures, This paper has been submitted to Automatica and the current publication decision is Accept provisionally as Brief Pape